Consider survey data from surgeries. $Y$ represents observed surgical quality and is measured post-surgery; $X$ represents perceived surgical difficulty level and is measured pre and post surgery.
It is desired to assess the relationship between $Y$ and $X_{pre}$ and also $Y$
on $X_{\delta} = X_{post} - X_{pre}$. However, since $X_\delta$ is derived from
$X_{pre}$, we know that $X_{pre}$ and $X_{\delta}$ will be highly correlated. One
option is to attempt to reduce such multicollinearity via centering.
Any thoughts on alternative strategies and pros/cons? This scenario
sounds similar to the commonly discussed scenario of how to handle
change scores via ANCOVA (e.g., Senn 2009, Stat Med) when we have change
scores with respect to $Y$, but it is different since here we have no
baseline $Y$ and a change score for $X$.
